How to plot a function graph
Video: Graphing a Basic Function 2024, July
We draw pictures with mathematical meaning, or rather, learn to build graphs of functions. Consider the construction algorithm.
Instruction manual
one
Investigate the domain (permissible values of the argument x) and the range of values (admissible values of the function y (x) itself). The simplest restrictions are the presence of trigonometric functions, roots or fractions with a variable in the denominator in the expression.
2
See if the function is even or odd (that is, check its symmetry with respect to the coordinate axes) or periodic (in this case, the components of the graph will be repeated).
3
Investigate the zeros of the function, that is, the intersections with the coordinate axes: if there are any, and if so, mark the characteristic points on the graph blank, and also examine the intervals of sign constancy.
4
Find the asymptotes of the graph of the function, vertical and inclined.
To find the vertical asymptotes, we study the discontinuity points on the left and right; to find the inclined asymptotes, the limit separately for plus infinity and minus infinity is the ratio of the function to x, that is, the limit on f (x) / x. If it is finite, then this is the coefficient k from the tangent equation (y = kx + b). To find b, you need to find the limit at infinity in the same direction (that is, if k is at plus infinity, then b is at plus infinity) of the difference (f (x) -kx). Substitute b into the equation of the tangent. If k or b could not be found, that is, the limit is infinity or does not exist, then there are no asymptotes.
5
Find the first derivative of the function. Find the values of the function at the obtained extremum points, indicate the areas of monotonous increase / decrease of the function.
If f '(x)> 0 at each point of the interval (a, b), then the function f (x) increases on this interval.
If f '(x) <0 at each point of the interval (a, b), then the function f (x) decreases on this interval.
If the derivative, when passing through the point x0, changes its sign from plus to minus, then x0 is the maximum point.
If the derivative, when passing through the point x0, changes its sign from minus to plus, then x0 is the minimum point.
6
Find the second derivative, that is, the first derivative of the first derivative.
It will show the bulge / concavity and inflection points. Find function values at inflection points.
If f "(x)> 0 at each point of the interval (a, b), then the function f (x) will be concave on this interval.
If f "(x) <0 at each point of the interval (a, b), then the function f (x) will be convex on this interval.
Useful advice
It is possible to make several intermediate images for construction, in order to avoid confusion and loss of some data and marks on the chart blank